Quantum groups and integrable systems

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Objective

The theory of quantum groups was created some 10 years ago in relation to problems arising in theoretical physics, such as conformal field theory, inverse scattering method, etc. Its mathematical foundation is a mixture of algebra (theory of groups or Hopf algebras) and differential geometry (Poisson brackets on manifolds or Lie groups). Since then it has found a large number of connections with or applications to other scientific domains, such as classification of knots, representation of finite groups or integrable systems.

The project will develop further the notion of an integrable system which, briefly, is a system of differential equations possessing sufficiently many first integrals. The theory of quantum groups is expected to throw more light on various problems such as the quantization of the classical situation (that is, the passage from classical to quantum mechanics) and the study of bi-Hamiltonian manifolds (manifolds endowed with two Poisson brackets).